In many practical situations, we are faced with the problem of recovering a signal that...

In many practical situations, we are faced with the problem of recovering a signal that has been “blurred” by a convolution process. We can model this blurring process as a linear filtering operation, as depicted in Figure P5.65-1, where the blurring impulse response is as shown in Figure P5.65-2. This problem will consider ways to recover x[n] from y[n].

(a) One approach to recovering x[n] from y[n] is to use an inverse filter; i.e., y[n] is filtered by a system whose frequency response is

where H(e^jω) is the Fourier transform of h[n]. For the impulse response h[n] shown in Figure P5.65-2, discuss the practical problems involved in implementing the inverse filtering approach. Be complete, but also be brief and to the point.

(b) Because of the difficulties involved in inverse filtering, the following approach is suggested for recovering x[n] from y[n]: The blurred signal y[n] is processed by the system shown in Figure P5.65-3, which produces an output w[n] from which we can extract an improved replica of x[n]. The impulse responses h₁[n] and h₂[n] are shown in Figure P5.65-4. Explain in detail the working of this system. In particular, state precisely the conditions under which we can recover x[n] exactly from w[n]. Hint: Consider the impulse response of the overall system from x[n] to w[n].

(c) Let us now attempt to generalize this approach to arbitrary finite-length blurring impulse responses h[n]; i.e., assume only that h[n] = 0 for n < 0 or n ≥ M. Further, assume that h₁[n] is the same as in Figure P5.65-4. How must H₂(z) and H(z) be related for the system to work as in part (b)?What condition must H(z) satisfy in order that it be possible to implement H₂(z) as a causal system?